- Email: [email protected]
Uncertainty evaluation of displacements measured by electronic speckle-pattern interferometry
Optics Communications 241 (2004) 279–292 www.elsevier.com/locate/optcom
Uncertainty evaluation of displacements measured by electronic speckle-pattern interferometry Rau´l R. Cordero
a,b,*
, Amalia Martı´nez c, Ramo´n Rodrı´guez-Vera c, Pedro Roth d
a
d
Faculty of Mechanical and Production Sciences Engineering, Escuela Superior Polite´cnica del Litoral, Km. 30, 5 Via Perimetral, Guayaquil, Ecuador b Department of Mechanical and Metallurgical Engineering, Pontificia Universidad Cato´lica de Chile, Vicun˜a Mackenna 4860, Santiago, Chile c ´ ptica, Apartado Postal 1-948, C.P. 37000, Leo´n, Gto., Me´xico Centro de Investigaciones en O Department of Mechanical Engineering, Universidad Te´cnica Federico Santa Maria, Ave. Espan˜a 1680, Valparaı´so, Chile Received 31 March 2004; received in revised form 6 July 2004; accepted 16 July 2004
Abstract We have applied electronic speckle-pattern interferometry (ESPI), a whole-field optical technique, to measure the displacements induced by applying tensile load on a metallic sheet sample. Because we used a dual-beam ESPI interferometer with collimated incident beams, our measurements were affected by errors in the collimation and in the alignment of the illuminating beams of the optical setup. In this paper, the influences of these errors are characterized and compared with other systematic effects through an uncertainty analysis. We found that the displacement uncertainty depends strongly on the incidence angles of the illuminating beams of the interferometer. Moreover, faults in the alignment of the incident beams have more influence on the uncertainty than errors in their collimation. The latter errors change the incident beams from collimated to slightly divergent, modifying in turn the interferometer sensitivity. We found that this sensitivity change can be generally neglected. 2004 Elsevier B.V. All rights reserved. PACS: 42.30.M; 42.40.K; 07.60.L Keywords: Speckle interferometry; Displacement measurements; Uncertainty analysis
1. Introduction *
Corresponding author. Tel.: +5632654501; +5632797656. E-mail address: [email protected] (R.R. Cordero).
fax:
Electronic speckle-pattern interferometry (ESPI) [1] and Moire´ interferometry [2] are used to obtain
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.07.040
280
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
relative displacement fields from fringe patterns that may be interpreted as contour maps of the phase difference induced by the specimen deformation. The automatic extraction of the information encoded in a pattern, includes several stages: first, phase-shifting technique [3] or Fourier transform method [4] is used to obtain the wrapped phase map; second, phase unwrapping is performed to obtain the phase differences induced by the deformation [5]; third, the relative displacements are evaluated by using the unwrapped phase difference values and the adequate sensitivity vector components. Independently of the phase measuring technique or the interferometer used, displacement measurements are affected by several random and systematic influences. These include: optical noise, environmental perturbations, characteristics of the phase measuring technique, misalignments and collimation faults of the beams, etc. The measured values of the phase are mainly affected by optical noise and environmental perturbations. In order to minimize and to compensate these problems, efforts have been done in the determination of the influence of noise under some particular conditions [6] and in the experimental quantification of the effect of the environmental vibration on Moire´ interferometry [7]. Moreover, some systematic errors in the measured phase, introduced by limitations in the phase measuring and unwrapping techniques, have been reported [8–12]. The displacement measurements depend also on the sensitivity vector of the interferometer. If collimated wavefronts are used, a dual-beam interferometer can be set up with sensitivity along just one spatial coordinate; this means that two components of the sensitivity vector are nominally zero. Alternatively, if one uses an interferometer with divergent illumination, it inevitably has sensitivity along all the spatial coordinates [13–15]. If the illuminating beams are collimated, sensitivity vector components are affected by errors in the collimation and in the alignment of the beams. The collimation errors produce slightly divergent beams, hence changing the sensitivity of the interferometer. Although, it has been reported a comparison between the components of the sensitivity vectors of ESPI interferometers that
use either collimated or spherical illumination [16], the problem of the uncertainty evaluation of displacement measured by an interferometer with nominally collimated illumination remains open. In this work, we use internationally accepted recommendations [17] to evaluate the uncertainty of the relative displacements measured by a symmetrical dual-beam ESPI interferometer that uses collimated wavefronts. The displacements were induced by applying tensile load to an aluminum sheet sample. In our analysis, special attention was paid to include the uncertainty sources associated to the change of the interferometer sensitivity caused by faults in the collimation. We found that the uncertainty of the components of the sensitivity vector depend strongly on the incidence angles, measured with respect to the specimen normal. Furthermore, we established that the errors in the alignment had more influence on the uncertainty of the displacements, than the faults in the collimation. Although the uncertainties of the nominally null components of the sensitivity vector were not insignificant, their contributions to the displacement uncertainty were negligible. Therefore, we conclude that the changes of the interferometer sensitivity caused by faults in the collimation can be ignored.
2. Experimental details Speckle-based methods, such as ESPI, are based on the digital subtraction of two video frames captured meanwhile a sample undergoes mechanical deformation. We used a dual-beam interferometer, with two mutually coherent collimated beams impinging from opposite sides upon a rough specimen surface Lx · Ly of 30 mm per side (Fig. 1). The sample was an 1100 aluminum sheet sample, 0.58 mm thick in the ‘‘as received’’ state (i.e., cold worked) and cut along the rolling direction. A first scattered speckle pattern was captured by a CCD camera of 512 · 512 pixels focused on the sample surface. A second pattern was recorded after the deformation of the specimen. The deformation was induced by an Instron machine working in tension along x. The digital subtraction of these patterns yielded the ESPI fringe pattern shown in Fig. 2(a). Conventional phase-shifting technique
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
281
Fig. 1. ESPI optical setup with sensitivity in the x-direction. The incidence angles of the beams were c = 49.4.
Fig. 2. (a) Fringe pattern obtained with the ESPI interferometer of Fig. 1. (b) Local relative displacements over y = 0 evaluated from pattern of (a).
of four frames with p/2 phase steps [3] was used to determine the whole-field phase-difference D/ between the captured patterns. According to [18], the induced displacement vector d = (U, V, W)T, is related with the phase-difference D/ through
D/ ¼ d e;
ð1Þ
where e = (ex, ey, ez)T is the sensitivity vector of the interferometer. Since our interferometer used collimated incident beams traveling in the plane x–z, according to [18], the components of the sensitivity
282
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
vector were, ey = 0, ez = 0 and ex = 4psen c/k, where k is the wavelength of the laser light and c is the incidence angle of each beam (measured with respect to the sample normal). It should be observed that, for this interferometer, two sensitivity vector components were nominally zero. Therefore, according to Eq. (1), we evaluated the estimates of the relative displacements along x, by U¼
kD/ ; 4psen c
ð2Þ
with k = 0.6328 lm and c = 49.4. The experiment was carried out in an optical metrology laboratory under constant temperature (20 C) and controlled conditions of dust and air currents. The displacements were calculated relative to the sample center where the fringe order zero was assigned. Conveniently, the origin of spatial reference system, x–y, was also located at the center of the field. Since the local displacements are proportional to the fringe orders at each field point (x, y), by observing the pattern shown in Fig. 2(a), we concluded that the displacement values increased in the x-direction and that the most strained area of the field was that close to center of the field, where the fringes appear closely spaced. The local relative displacements over y = 0, evaluated by Eq. (2), are plotted in Fig. 2(b). It can be observed that the local displacement increased almost linearly with x. This means that the strain (oU/ox) can be considered practically constant along x.
3. Law of propagation of uncertainties Consider a vector of input quantities p = (p1 pn)T related to an unknown vector of output quantities q = (q1 qm)T through a set of measurement models M(p, q) = 0, where 0 is a null m-dimensional vector. The n · n symmetric input uncertainty (or covariance) matrix is assumed to be known. It is expressed as: 2 2 3 u ðp1 Þ uðp1 ; pn Þ 6 7 .. .. .. 7; u2p ¼ 6 ð3Þ . . . 4 5 uðp1 ; pn Þ
u2 ðpn Þ
where the diagonal terms are the squares of the standard uncertainties of the input quantities and the off-diagonal terms are their mutual uncertainties. The latter terms are zero if the quantities are uncorrelated. The output values are obtained by solving the measurement models. If these models are linear or weakly nonlinear, the m · m output uncertainty matrix u2q is obtained by applying the so-called generalized law of propagation of uncertainties (GLPU). According to [19], this law is expressed as u2q ¼ S u2p ST ;
ð4Þ
where S is the m · n global sensitivity matrix: S ¼ ðSq Þ
1
Sp ;
ð5Þ
and Sq and Sp are, respectively, the m · m output and m · n input sensitivity matrices: 2 oM
1
oq1
6 6 Sq ¼ 6 ... 4
..
oM m oq1
2 oM
1
op1
6 6 Sp ¼ 6 ... 4
oM m op1
.
oM 1 oqm
7 .. 7 ; . 7 5
oM m oqm
oM 1 opn
..
.
3 ð6Þ
3
7 .. 7 : . 7 5
ð7Þ
oM m opn
As an example, we have applied the GLPU to the simple model z f(x,y) = 0. Following the formulation established above, in this case the vector of the input quantities is P = (x,y)T and the output quantity is Q = (z). Therefore n = 2 and m = 1. Additionally, the input uncertainty matrix, formed by the standards and the mutual uncertainties of x and y, is known: u2p
¼
u2 ðxÞ
uðx; yÞ
uðx; yÞ
u2 ðyÞ
:
ð8Þ
Applying Eqs. (6) and (7) we obtained that Sq = (1) and that Sp = (of/ox of/oy)T. Therefore by applying Eq. (4), we achieved the so-called law of propagation of uncertainties (LPU) for two input quantities:
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
u2 ðzÞ ¼
2 2 of of u2 ðxÞ þ u2 ðyÞ ox oy of of þ2 uðx; yÞ: ox oy
283
ommended using a uniform pdf [17,19,20]. Then, its standard uncertainty is taken as ð9Þ
The generalization of this law to more than two input quantities is straightforward. The terms (of/ ox)2u2(x) and (of/oy)2u2(y) are the contributions of the input quantities x and y, to the square of the uncertainty of the output quantity z. It should be observed that in the case of a single output quantity, the GLPU reduces to the LPU, and the matrix formulation becomes unnecessary. There are two approaches to evaluate the standard uncertainties of the input quantities (diagonal elements of the input uncertainty matrix). The type A evaluation [17] applies only to quantities which are measured directly several times under repeatability conditions. Uncertainty of input quantities that are measured only once, those are evaluated from models that involve further quantities, or that are imported from other sources, should be evaluated by the type B method of evaluation [17]. In many cases, this type involves obtaining an uncertainty as the standard deviation of the probability density function (pdf) that is assumed to apply. For example, if a quantity X is assumed to vary uniformly within a given range of width dX, is rec-
dX uðX Þ ¼ pffiffiffiffiffi : 12
ð10Þ
4. Sensitivity vector uncertainty 4.1. Sensitivity vector models According to [16], the sensitivity vector e = (ex, ey, ez)T of a dual-beam interferometer can be expressed as e¼
2p ½^n1 ^n2 ; k
ð11Þ
where the unitary vectors n^1 and ^n2 stand for the direction of the illuminating rays on each point of the observed area. The vectors ^n1 , ^n2 and e are shown in Fig. 3. The vector ^n1 can be expressed through a model that includes the angles a1, b1 and c1 of this unitary vector with respect to the corresponding coordinates x, y and z (see Fig. 3). ^ where ^n1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hence, p ¼ cos a1^i þ cos bffi 1^j cos c1 k, cos a1 ¼ 1 cos2 b1 cos2 c1 . If the angles of ^n2 with respect to the coordinates x, y and z are, respectively, a2, b2 andc2 according to Eq. (11),
Fig. 3. Sensitivity vector e of a dual-beam interferometer.
284
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
the components of the sensitivity vector e of Fig. 3 can be written as ex ¼
ffi 2p hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 cos2 ðc1 þ Dc1 Þ cos2 ðb1 þ Db1 Þ k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii þ 1 cos2 ðc2 þ Dc2 Þ cos2 ðb2 þ Db2 Þ ; ð12aÞ
ey ¼
2p ½cos ðb1 þ Db1 Þ cos ðb2 þ Db2 Þ; k
ð12bÞ
ez ¼
2p ½ cos ðc1 þ Dc1 Þ þ cos ðc2 þ Dc2 Þ; k
ð12cÞ
where we included the quantities Dc1, Dc2 and Db2 to account for eventual errors in the collimation of the incident beams. Although in our experiment the estimates of Dc1, Dc2, Db1 and b2 were zero, their corresponding uncertainties were not. Since we used the symmetrical interferometer of Fig. 1, with collimated beams traveling in the x–z plane, we considered that b1 = b2 = 90 and that c1 = c2. Therefore, by evaluating Eqs. (12a–c) we obtained that the estimates of the components of the sensitivity vector of this interferometer were ex = 4p sen c1/k, ey = 0 and ez = 0. Estimates of the sensitivity vector components can be affected by errors in the collimation and errors in the alignment of the wavefronts used to illuminate the sample. The latter errors can shift the values of b1, b2, c1 and c2; and the former errors can affect the values of Dc1, Dc2, Db1 and Db2. If Dc1, Dc2, Db1 and Db2 are not zero, ey and ez, evaluated by Eqs. (12b–c) are not zero. This means that the collimation errors produce slightly divergent beams changing in turn the sensitivity of the interferometer. 4.2. Uncertainty of the input quantities According to the formulation established in Section 3, with Eqs. (12a–c) we built up a set of three measurement models that we represented compactly as M(p, e) = 0, where the 8-dimensional vector of input quantities p = (b1, b2, c1, c2, Dc1, Dc2, Db1, Db2)T is related to the vector of output
quantities e = (ex, ey, ez)T. Since the maximum reasonable error associated to the estimated value of k is just about 0.1 nm, we decided to neglect the contribution of k to the uncertainty of the sensitivity vector components. Therefore, k was not considered as an element of the input vector p. Because, we assumed that the input quantities were uncorrelated, the 8 · 8 input uncertainty matrix u2p was diagonal. Its diagonal terms were the squares of the standard uncertainties of the input quantities b1, b2, c1, c2, Dc1, Dc2, Db1 and Db2. c1 is the angle with respect to z of the symmetry axis of the beam 1 of the interferometer of Fig. 3. Its standard uncertainty was taken as equal to the standard deviation of the uniform probability density function (pdf) that we assumed to apply. The width of the pdf was assigned considering the maximum reasonable deviation of c1 with respect to its estimated value. This deviation is the alignment error. Thus, according to Eq. (10) and considering that the maximum reasonable alignment error was 1, we used 2ð 1 Þ uðc1 Þ ¼ pffiffiffiffiffi : 12
ð13Þ
Analogous equations to Eq. (13) were utilized to evaluate the uncertainties of c2, b1 and b2, considering for these input quantities the same maximum reasonable alignment error. Fig. 4(a) shows the transversal section of the beam 1 of the interferometer of Fig. 3. This impinges on an area Lx · Ly and it has collimation errors. Since ^n1 stands for the direction of the illuminating ray on each point of the observed area, on the plane x–z, the collimation error Dc1 was defined as the angle between the symmetry axis of the beam and the local vector ^n1 . Similarly, on the plane y–z, Db1 was the angle between the symmetry axis of the beam and the local vector ^n1 . As it is shown in Fig. 4(b), the collimation errors depend on the position (x, y) of the illuminated point. At point (Lx/2, Ly/2), the corresponding illuminating ray has the largest collimation errors. For any other point (x, y) of the illuminated surface, we estimated these errors as x Dc1;Lx Dc1 ¼ ; ð14aÞ ðLx =2Þ
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
285
Fig. 4. (a) Transversal section of the beam 1 of Fig. 3, impinging on an area Lx · Ly. This beam has collimation errors and therefore, it appears slightly divergent. The collimation errors at the borders are denoted as Dc1,Lx and Db1,Ly. (b) Change in the collimation error with the position of the illuminated point. It should be observed that Dc1,Lx > Dc1.
y Db1;Ly ; Db1 ¼ Ly =2
ð14bÞ
where Dc1,Lx and Db1,Ly stand for the collimation errors of the rays that illuminate the borders of the observed surface. During our experiment, we presumed an adequate collimation and therefore the estimates of Dc1, Dc2, Db1 and Db2 were zero. However, their uncertainties were not. The standard uncertainties of Dc1 and Db1 were obtained by Eq. (10) assuming uniform pdfs. The width of the pdf was determined considering the maximum reasonable deviations of Dc1 and of Db1 with respect to their corresponding estimates. These deviations were evaluated considering for the angles Dc1,Lx and Db1,Ly in Eqs. (14a–b), values of 0.1. Then, since Lx and Ly were both equal to 30 mm, we obtained
uðDc1 Þ ¼
2xð0:1Þ pffiffiffiffiffi =mm; 15 12
uðDb1 Þ ¼
2y ð0:1Þ pffiffiffiffiffi =mm: 15 12
ð15aÞ
ð15bÞ
Analogous equations to Eqs. (14a–b) and to Eqs. (15a–b) were utilized to evaluate the uncertainties of Dc2 and for Db2, considering for Dc2,Lx and for Db2,Ly, angles of 0.1.
4.3. Uncertainty of the components of e According to the formulation established in Section 3, we built up a 8 · 8 diagonal input uncertainty matrix u2p with the standard uncertainties of the input quantities (elements of the vector p = (b1, b2, c1, c2, Dc1, Dc2, Db1, Db2)T). The output quantities (elements of the vector e = (ex, ey, ez)T) were related to the input quantities through Eqs. (12a–c) that formed a set of three measurement models written compactly as M(p, e) = 0. Then, by applying the GLPU (Eq. (4)), we obtained the 3 · 3 output uncertainty matrix 2 2 3 u ðex Þ uðex ; ey Þ uðex ; ez Þ 6 7 u2e ¼ 4 uðex ; ey Þ u2 ðey Þ uðez ; ey Þ 5: ð16Þ uðex ; ez Þ
uðez ; ey Þ
u2 ðez Þ
The diagonal terms of this matrix are the squares of the standard uncertainties of the components of the sensitivity vector e, and the off-diagonal terms are their mutual uncertainties. The application of the GLPU yielded zero for the terms u(ex, ey) and u(ez, ey) of the output uncertainty matrix u2e . Fig. 5 shows the obtained whole-field values of u(ex), u(ey), u(ez), and u(ex, ez). It can be observed that, although the uncertainties of ey and ez are not negligible, the mutual uncertainty
286
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
Fig. 5. Standard uncertainty of: (a) ex, (b) ey and (c) ez and (d) mutual uncertainty of ex and ez.
of ex and ez is very small and therefore, we considered that it can be ignored. The relative standard uncertainty of ex, calculated as, along line y = 1 mm is shown in Fig. 6. It can be observed that the relative standard uncertainty of ex calculated using the incidence angles (c1 and c2) both equal to 49.4 was about 0.6%. It is useful to compare this datum with that evaluated for a Moire´ interferometer. Incidence angles of 49.4 are used in Moire´ interferometry to illuminate a grating of 1200 lines/mm that can be glued onto the surface of a sample. According to the data reported by [7], the relative standard uncertainty of ex of a Moire´ interferometer that uses a commercial grating of 1200 lines/mm, is about 0.4%. In Fig. 6, the relative standard uncertainties of ex calculated using the same data and input uncertainties but changing the incidence angles of the beams (c1 and c2) measured with respect to the specimen normal are also shown. It can be observed that the standard uncertainty of ex de-
Fig. 6. Relative standard uncertainty of ex along y = mm, calculated using for both incidence angles (c1 and c2) the values indicated in the plot.
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
pended strongly on the angles of the incident wavefronts. Moreover, the spatial variation of this uncertainty along line y = 1 mm was small. 4.4. Nonlinearity problems Since Eq. (12a) and (12b) use some common quantities (b1, b2, Db1 and Db1), ex and ey are correlated. However, the mutual uncertainty u(ex, ey), obtained in Section 4.3 by applying the GLPU, was zero. This was not a reasonable result and it was due to the strong non-linearity of Eqs. (12a– c). Nevertheless, since the mutual uncertainty of ex and ez was small (see Fig. 5(d)), the mutual uncertainty of ex and ey was also presumed small and therefore we considered that the correlation of the components of the sensitivity vector can be ignored. Another nonlinearity problem was that linked with the terms cos2(b1 + Db1) and cos2(b2 + Db2) in Eq. (12a). Since the estimates of b1 andb2 were 90 and the estimates of Db1 and Db2 were zero, when the GLPU was applied, the contributions of b1, b2, Db1 and Db2 to the uncertainty of ex were zero irrespective of the uncertainties associated to these input quantities. This was not a reasonable result, and it was due to the strong nonlinearity of the square of the cosine function in the vicinity of 90. In order to evaluate the lost contributions of the quantities b1, b2, Db1 and Db2, to the standard uncertainty of ex, we decided to consider the term cos2(b1 + Db1) in Eq. (12a) as a single input quantity that we labeled as S1. Similarly, S2 ” cos2(b2 + Db2). In order to evaluate the standard uncertainty of the new input quantity S1, first, we reduced the term cos2(b1 + Db1): S 1 cos2 ðb1 þ Db1 Þ ¼ 12½1 þ R1 ;
287
tively, at b1 = 90 and at Db1 = 0. The widths of these pdfs were determined considering the maximum reasonable deviation angles with respect to their estimates. For Db1, this deviation was estimated using Eq. (14b); it gave 0.006.y. For b1, we used 1. Therefore 1 1 E½R21 ¼ 0:012 :y 2 Z 0006 :y Z 91 cos2 ð2b1 þ 2Db1 Þ db1 dDb1 ; 0006 :y
89
ð19aÞ 1 1 E½R1 ¼ 0:012 :y 2 Z 0006 :y Z 91 cos ð2b1 þ 2Db1 Þ db1 dDb1 :
0006 :y
89
ð19bÞ E½R21
and E[R1] evaluated by Eqs. (19a) and (19b) allowed us to estimate u2(R1) by Eq. (18). Since Eq. (17) is a single measurement model, the matrix formulation of Section 3 was unnecessary and the standard uncertainty of S1 was calculated by applying the LPU (Eq. (9–17)). The results are shown in Fig. 7 and they can be considered also valid for. Although the scale of the plot was selected in order to show the spatial variations of u(S1), these variations were relatively small. Computing the spatial average of the data plotted in Fig. 7, we concluded that the standard uncertainty
ð17Þ
where R1 = cos(2b1 + 2Db1). The application of the law of propagation of uncertainties to Eq. (17) allowed us to evaluate the uncertainty of S1 which in turn depended on the uncertainty of R1. According to the procedure recommended by [19], the latter was evaluated by 2
u2 ðR1 Þ ¼ E½R21 ½E½R1 ;
ð18Þ
where E is the expectation operator. We assumed for b1 and for Db1 uniform pdfs centered, respec-
Fig. 7. Standard uncertainty of S1 ” cos2(b1 + Db1).
288
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
of was just about 9.2 · 105. This means that the combined contribution of the input quantities b1 and Db1 to the standard uncertainty of ex should be very small. The same is valid for the contributions of b2 and Db2. Therefore, we concluded that the uncertainty evaluation performed in Section 4.3 was not affected significantly by the application of the GLPU to the non-linear equations (12a–c). Nevertheless, the values of u(ex) shown in Fig. 5(a) include the combined contribution of b1, Db1, b2 and Db2, calculated through the evaluation of u(S1) and u(S2) following the procedure described above. 4.5. Contributions to the ex uncertainty Fig. 8 depicts the different contributions to the square of u(ex) along line y = 1 mm, as a
function of the square of x. For clarity, these contributions have been labeled as C2(X) ” (oex/ oX)2u2(X) for generic input quantity X. The scale of the plot was selected in order to show the very small contributions of and, that in turn stand for the combined contributions of b1, Db1, b2 and Db2. It may be seen that the contributions of c1 and c2 are greater than those of Dc1 and Dc2. This means that eventual errors in the alignment have more influence on the uncertainty than faults in the collimation. Therefore, we conclude that, if a dual-beam interferometer is used, special attention must be paid to the accurate alignment of the incident beams. Moreover, since in Fig. 8 the contributions of Dc1 and Dc2 increase with the spatial coordinate x, we conclude that the effect of the collimation errors was greater on the illuminated area close to the borders. This means that the relative influence of the collimation errors on the uncertainty should increase with the size of the field.
5. Displacement uncertainty 5.1. Displacement model During a mechanical tensile test with load application along x, although the main relative displacement of the specimen should be observed along that direction, the sample dimension change along x induces also deformations along coordinates y and z. In this case, the three normal strains can be related by using the Poisson rate v [21]: ey ¼ ez ¼ vex ;
Fig. 8. Contributions to u2(ex) as a function of x2, along y = 1 mm. Line 1: C2(c1) + C2(c2); Line 2: C2(c1) + C2(c2) + C2[S1] + C2[S2]; Line 3: C2(c1) + C2(c2) + C2[S1] + C2[S2] + C2(Dc1) + C2(Dc2).
ð20Þ
where ex = oU/ox, ey = oV/oy and ez = oW/oz. As it can be observed in Fig. 2(a), we located both, the reference system utilized to determine the spatial coordinates of the field and the reference system used to evaluate the displacements, at the center of the illuminated area. Moreover, Fig. 2(b) has shown that the local displacement increased almost linearly with x. Therefore, for values of x and y different of zero, the strains associated to the local displacement components could be approximately evaluated by:
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
ex ¼
U ; x
ð21aÞ
ey ¼
V ; y
ð21bÞ
ez ¼
WT ; Lz
ð21cÞ
where, the total displacement induced along z(WT) and the sample thick (Lz) were used to evaluate ez. Combining Eqs. (1) and (20) and Eqs. (21a–c), and solving for U, we obtain U¼
D/x : ex x vey y vez Lz
ð22Þ
Since the components of the sensitivity vector of our ESPI interferometer were ex = 4p sen c1/k, ey = 0 and ez = 0, Eq. (22) reduces to Eq. (2). However, because the uncertainties of ey and ez were not negligible (see Fig. 5), we considered that Eq. (2) does not include all the uncertainty sources. Therefore, we took Eq. (22) as the measurement model to evaluate the displacement uncertainty by applying the law of propagation of uncertainties. 5.2. Uncertainty of the input quantities According to the formulation established in Section 3, Eq. (22) is a single measurement model where U is related to the input quantities D/, ex, ey and ez. Since the estimates of ey and ez were zero, when the LPU (Eq. (9)) was applied to Eq. (22), the contributions to the U uncertainty of the terms v and Lz were zero. Although it has been recommended several methods to include these lost contributions [19], independently of the selected technique, these contributions are very small compared to those of D/ and ex. Therefore, we decided ignore the standard uncertainties of v and Lz ; they were not considered as input quantities. As pointed out in Section 4.4, although the components of the sensitivity vector are correlated, in our case, their mutual uncertainties can be ignored. Therefore, we applied the law of propagation of uncertainties to the model (22) assuming that the input quantities were uncorrelated.
289
The standard uncertainty of D/ depends on systematic effects linked with the used phase-shifting procedure and on the influence of noise and the perturbing environment. Assuming that the first one was compensated adequately, we have considered just the eventual variations in the phase-difference caused by the optical noise and the external influence. We estimated a maximum eventual error of about 0.5 rad in the measured D/ values. Therefore, assuming a uniform pdf, we took 2ð0:5Þ uðD/Þ ¼ pffiffiffiffiffi rad: 12
ð23Þ
The standard uncertainty of D/, estimated by Eq. (23), was 0.28 rad. This is a standard uncertainty considerably greater than that reported for the D/ values measured under similar experimental conditions with a Moire´ interferometer of the same sensitivity of our ESPI interferometer. According to [7], for the D/ values measured by phase-shifting Moire´ interferometry, u( D/) is just about 0.13 rad. The difference can be explained by considering the different signal-to-noise ratios of ESPI and of Moire´ interferometry. Since ESPI yields patterns noisier than those obtained by Moire´, we speculate that in the former method, the influence of optical noise should be greater than in the case of Moire´. The standard uncertainties of ex,ey and ez were evaluated in Section 4.3. Fig. 5 shows these standard uncertainties. 5.3. Standard uncertainty of the displacement The standard uncertainty of U, at each point of the illuminated field, was calculated by applying the LPU (Eq. (9–22)). We used as input uncertainties the data plotted in Fig. 5, and the outcome of Eq. (23). Fig. 9 shows the whole-field results. Although the scale of the plot was selected in order to shown the spatial variations of the local displacement uncertainties, these variations were relatively small. Computing the spatial average of the data plotted in Fig. 9, we concluded that the displacement uncertainty associated to our measurements was about 0.019 lm. The uncertainties associated to the displacement measurements performed by an ESPI
290
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
Fig. 9. (a) Standard uncertainty map of U. (b) Contour diagram of (a).
interferometer are considerably greater than those reported for similar displacements when a Moire´ interferometer of the same sensitivity was used. Based on the data reported by [7], the uncertainty associated to a relative displacement of 1 lm, measured by using a Moire´ interferometer, is about 0.009 lm. The difference observed when it is compared this datum with the uncertainties plotted in Fig. 9, can be explained considering that the influence of the optical noise on speckle-based methods is greater than on Moire´ techniques. 5.4. Contributions to the displacement uncertainty Fig. 10 depicts the different contributions to the square of u(U) along line x = 1 mm, as a function of the square of U, where again C2(X) denotes the contribution of generic quantity X. It may be seen that the most important contributor to the displacement standard uncertainty was the corresponding phase-difference. Since the closely linear relationship between the displacement and the spatial coordinate x (see Fig. 2(b)), the contribution of ex increased almost linearly with the squares of the corresponding displacements. Although, as shown above, the uncertainties of ey and ez were not negligible, their contributions to the displacement uncertainty were imperceptible. This means that the changes of the nominally null sensitivity vector
Fig. 10. Contributions to u2(U) along y = 1 mm, as a function of U2 Line 1: C2(D/) Line 2: C2(D/) + C2(ex).
components, caused by faults in the collimation, can be ignored. Hence, since the expected errors in metrological applications of the speckle-based interferometric
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
techniques are generally smaller than the assumed maximum reasonable errors in the alignment and in the collimation, we concluded that reliable uncertainty evaluations of measurements obtained by an ESPI interferometer with collimated illumination, can be performed by applying the LPU to the simple model: U¼
kD/ : 2p½senðc1 þ Dc1 Þ þ senðc2 þ Dc2 Þ
ð24Þ
291
displacement uncertainty can be neglected if the maximum reasonable errors in alignment and in collimation are in the order of those assumed in this work. Since the expected errors in metrological applications of the speckle-based interferometric techniques are generally smaller than the assumed maximum reasonable errors, we conclude that the changes of the ESPI interferometer sensitivity, caused by faults in the collimation, can be generally ignored.
6. Summary and conclusions
Acknowledgements
An investigation was carried out to evaluate the uncertainty of displacements measured by using a dual-beam ESPI interferometer that used collimated illuminating beams. Displacements were induced by applying tensile load to a metallic sheet sample. The interferometer was sensible just along the pulling direction. The other two sensitivity vector components were nominally zero. Displacement measurements depend on the interferometer sensitivity that is affected by errors in the collimation and in the alignment of the illuminating beams. Collimation errors produce slightly divergent beams changing in turn the interferometer sensitivity. In this work, special attention was paid to evaluate the contributions to the displacement uncertainty of eventual changes of the interferometer sensitivity produced by faults in the collimation of the beams. We found that the uncertainty of the sensitivity vector components depended strongly on the angle of the incident wavefronts, measured with respect to the specimen normal. We established that the errors in the alignment had more influence on the uncertainty evaluation than the faults in the collimation. Moreover, we found that the effect of the collimation errors was greater on the illuminated area close to the borders. This means that the relative influence of the collimation errors on the uncertainty should increase with the size of the field. Although the uncertainties of the nominally null sensitivity vector components were not insignificant, we found that their contributions to the
R.R. Cordero thanks support of MECESUP PUC/19903 Project and VLaamse Interuniversitaire Road (VLIR IUC, Componente 6). A. Martı´nez and R. Rodrı´guez-Vera thank Consejo Nacional de Ciencia y Tecnologı´a (CONACYT) and Consejo de Ciencia y Tecnologı´a del Estado de Guanajuato (CONCYTEG) for their suport to carry out this research. We thank Dr. Gonzalo Fuster, Dr. Luciano Laroze and the referee, for helpful suggestions to the final draft of this paper.
References [1] R. Jones, C. Wykes, Holographic and Speckle Interferometry, first ed., Cambridge University Press, Cambridge, 1983. [2] D. Post, B. Han, P. Ifju, High Sensitivity Moire´: Experimental Analysis for Mechanics and Materials, Springer, New York, 1994. [3] J.M. Huntley, in: P.K. Rastogi (Ed.), Digital Speckle Pattern Interferometry and Related Techniques, Wiley, Chichester, 2001. [4] M. Takeda, H. Ina, S. Kobayashi, J. Opt. Soc. Am. 72 (1) (1982) 156160. [5] D.W. Robinson, G.T. Reid, Phase Unwrapping Methods. Interferogram Analysis, Institute of Physics, Bristol, UK, 1993. [6] M.R. Miller, I. Mohammed, P.S. Ho, Opt. Lasers Eng. 36 (2) (2001) 127. [7] R.R. Cordero, I. Lira, Opt. Commun. 237 (2004) 25. [8] J. Schwider, R. Burow, K.E. Elner, J. Grzanna, R. Spolaczyk, K. Merkel, Appl. Opt. 22 (1983) 3421. [9] J. Schmit, K. Creath, Appl. Opt. 34 (1995) 3610. [10] H. Zhang, M. Lalor, D. Burton, Opt. Lasers Eng. 31 (1999) 381. [11] K. Creath, J. Schmit, Opt. Laser Eng. 24 (1996) 365. [12] P.J. de Groot, L.L. Deck, Appl. Opt. 35 (1996) 2172.
292
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
[13] D. Albrecht, Opt. Lasers Eng. 31 (1) (1999) 6381. [14] H.J. Puga, R. Rodriguez-Vera, A. Martinez, Opt. Lasers Technol. 34 (1) (2002) 8192. [15] A. Martinez, R. Rodriguez-Vera, J.A. Rayas, H.J. Puga, Opt. Lasers Eng. 39 (2003) 525. [16] A. Martinez, R. Rodriguez-Vera, J.A. Rayas, H.J. Puga, Opt. Commun. 223 (2003) 239. [17] ISO Guide to the Expression of Uncertainty in Measurement, Geneva, 1993.
[18] T. Kreis, Holographic Interferometry Principles and Methods, first ed., Akademic, Berlin, 1996. [19] I. Lira, Evaluating the Uncertainty of Measurement: Fundamentals and Practical Guidance, Institute of Physics Publishing, Bristol, 2002. [20] R.R. Cordero, P. Roth, Metrologia 41 (4) (2004) L22. [21] S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, third ed., McGraw-Hill International Editions, Singapore, 1970.
Uncertainty evaluation of displacements measured by electronic speckle-pattern interferometry Rau´l R. Cordero
a,b,*
, Amalia Martı´nez c, Ramo´n Rodrı´guez-Vera c, Pedro Roth d
a
d
Faculty of Mechanical and Production Sciences Engineering, Escuela Superior Polite´cnica del Litoral, Km. 30, 5 Via Perimetral, Guayaquil, Ecuador b Department of Mechanical and Metallurgical Engineering, Pontificia Universidad Cato´lica de Chile, Vicun˜a Mackenna 4860, Santiago, Chile c ´ ptica, Apartado Postal 1-948, C.P. 37000, Leo´n, Gto., Me´xico Centro de Investigaciones en O Department of Mechanical Engineering, Universidad Te´cnica Federico Santa Maria, Ave. Espan˜a 1680, Valparaı´so, Chile Received 31 March 2004; received in revised form 6 July 2004; accepted 16 July 2004
Abstract We have applied electronic speckle-pattern interferometry (ESPI), a whole-field optical technique, to measure the displacements induced by applying tensile load on a metallic sheet sample. Because we used a dual-beam ESPI interferometer with collimated incident beams, our measurements were affected by errors in the collimation and in the alignment of the illuminating beams of the optical setup. In this paper, the influences of these errors are characterized and compared with other systematic effects through an uncertainty analysis. We found that the displacement uncertainty depends strongly on the incidence angles of the illuminating beams of the interferometer. Moreover, faults in the alignment of the incident beams have more influence on the uncertainty than errors in their collimation. The latter errors change the incident beams from collimated to slightly divergent, modifying in turn the interferometer sensitivity. We found that this sensitivity change can be generally neglected. 2004 Elsevier B.V. All rights reserved. PACS: 42.30.M; 42.40.K; 07.60.L Keywords: Speckle interferometry; Displacement measurements; Uncertainty analysis
1. Introduction *
Corresponding author. Tel.: +5632654501; +5632797656. E-mail address: [email protected] (R.R. Cordero).
fax:
Electronic speckle-pattern interferometry (ESPI) [1] and Moire´ interferometry [2] are used to obtain
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.07.040
280
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
relative displacement fields from fringe patterns that may be interpreted as contour maps of the phase difference induced by the specimen deformation. The automatic extraction of the information encoded in a pattern, includes several stages: first, phase-shifting technique [3] or Fourier transform method [4] is used to obtain the wrapped phase map; second, phase unwrapping is performed to obtain the phase differences induced by the deformation [5]; third, the relative displacements are evaluated by using the unwrapped phase difference values and the adequate sensitivity vector components. Independently of the phase measuring technique or the interferometer used, displacement measurements are affected by several random and systematic influences. These include: optical noise, environmental perturbations, characteristics of the phase measuring technique, misalignments and collimation faults of the beams, etc. The measured values of the phase are mainly affected by optical noise and environmental perturbations. In order to minimize and to compensate these problems, efforts have been done in the determination of the influence of noise under some particular conditions [6] and in the experimental quantification of the effect of the environmental vibration on Moire´ interferometry [7]. Moreover, some systematic errors in the measured phase, introduced by limitations in the phase measuring and unwrapping techniques, have been reported [8–12]. The displacement measurements depend also on the sensitivity vector of the interferometer. If collimated wavefronts are used, a dual-beam interferometer can be set up with sensitivity along just one spatial coordinate; this means that two components of the sensitivity vector are nominally zero. Alternatively, if one uses an interferometer with divergent illumination, it inevitably has sensitivity along all the spatial coordinates [13–15]. If the illuminating beams are collimated, sensitivity vector components are affected by errors in the collimation and in the alignment of the beams. The collimation errors produce slightly divergent beams, hence changing the sensitivity of the interferometer. Although, it has been reported a comparison between the components of the sensitivity vectors of ESPI interferometers that
use either collimated or spherical illumination [16], the problem of the uncertainty evaluation of displacement measured by an interferometer with nominally collimated illumination remains open. In this work, we use internationally accepted recommendations [17] to evaluate the uncertainty of the relative displacements measured by a symmetrical dual-beam ESPI interferometer that uses collimated wavefronts. The displacements were induced by applying tensile load to an aluminum sheet sample. In our analysis, special attention was paid to include the uncertainty sources associated to the change of the interferometer sensitivity caused by faults in the collimation. We found that the uncertainty of the components of the sensitivity vector depend strongly on the incidence angles, measured with respect to the specimen normal. Furthermore, we established that the errors in the alignment had more influence on the uncertainty of the displacements, than the faults in the collimation. Although the uncertainties of the nominally null components of the sensitivity vector were not insignificant, their contributions to the displacement uncertainty were negligible. Therefore, we conclude that the changes of the interferometer sensitivity caused by faults in the collimation can be ignored.
2. Experimental details Speckle-based methods, such as ESPI, are based on the digital subtraction of two video frames captured meanwhile a sample undergoes mechanical deformation. We used a dual-beam interferometer, with two mutually coherent collimated beams impinging from opposite sides upon a rough specimen surface Lx · Ly of 30 mm per side (Fig. 1). The sample was an 1100 aluminum sheet sample, 0.58 mm thick in the ‘‘as received’’ state (i.e., cold worked) and cut along the rolling direction. A first scattered speckle pattern was captured by a CCD camera of 512 · 512 pixels focused on the sample surface. A second pattern was recorded after the deformation of the specimen. The deformation was induced by an Instron machine working in tension along x. The digital subtraction of these patterns yielded the ESPI fringe pattern shown in Fig. 2(a). Conventional phase-shifting technique
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
281
Fig. 1. ESPI optical setup with sensitivity in the x-direction. The incidence angles of the beams were c = 49.4.
Fig. 2. (a) Fringe pattern obtained with the ESPI interferometer of Fig. 1. (b) Local relative displacements over y = 0 evaluated from pattern of (a).
of four frames with p/2 phase steps [3] was used to determine the whole-field phase-difference D/ between the captured patterns. According to [18], the induced displacement vector d = (U, V, W)T, is related with the phase-difference D/ through
D/ ¼ d e;
ð1Þ
where e = (ex, ey, ez)T is the sensitivity vector of the interferometer. Since our interferometer used collimated incident beams traveling in the plane x–z, according to [18], the components of the sensitivity
282
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
vector were, ey = 0, ez = 0 and ex = 4psen c/k, where k is the wavelength of the laser light and c is the incidence angle of each beam (measured with respect to the sample normal). It should be observed that, for this interferometer, two sensitivity vector components were nominally zero. Therefore, according to Eq. (1), we evaluated the estimates of the relative displacements along x, by U¼
kD/ ; 4psen c
ð2Þ
with k = 0.6328 lm and c = 49.4. The experiment was carried out in an optical metrology laboratory under constant temperature (20 C) and controlled conditions of dust and air currents. The displacements were calculated relative to the sample center where the fringe order zero was assigned. Conveniently, the origin of spatial reference system, x–y, was also located at the center of the field. Since the local displacements are proportional to the fringe orders at each field point (x, y), by observing the pattern shown in Fig. 2(a), we concluded that the displacement values increased in the x-direction and that the most strained area of the field was that close to center of the field, where the fringes appear closely spaced. The local relative displacements over y = 0, evaluated by Eq. (2), are plotted in Fig. 2(b). It can be observed that the local displacement increased almost linearly with x. This means that the strain (oU/ox) can be considered practically constant along x.
3. Law of propagation of uncertainties Consider a vector of input quantities p = (p1 pn)T related to an unknown vector of output quantities q = (q1 qm)T through a set of measurement models M(p, q) = 0, where 0 is a null m-dimensional vector. The n · n symmetric input uncertainty (or covariance) matrix is assumed to be known. It is expressed as: 2 2 3 u ðp1 Þ uðp1 ; pn Þ 6 7 .. .. .. 7; u2p ¼ 6 ð3Þ . . . 4 5 uðp1 ; pn Þ
u2 ðpn Þ
where the diagonal terms are the squares of the standard uncertainties of the input quantities and the off-diagonal terms are their mutual uncertainties. The latter terms are zero if the quantities are uncorrelated. The output values are obtained by solving the measurement models. If these models are linear or weakly nonlinear, the m · m output uncertainty matrix u2q is obtained by applying the so-called generalized law of propagation of uncertainties (GLPU). According to [19], this law is expressed as u2q ¼ S u2p ST ;
ð4Þ
where S is the m · n global sensitivity matrix: S ¼ ðSq Þ
1
Sp ;
ð5Þ
and Sq and Sp are, respectively, the m · m output and m · n input sensitivity matrices: 2 oM
1
oq1
6 6 Sq ¼ 6 ... 4
..
oM m oq1
2 oM
1
op1
6 6 Sp ¼ 6 ... 4
oM m op1
.
oM 1 oqm
7 .. 7 ; . 7 5
oM m oqm
oM 1 opn
..
.
3 ð6Þ
3
7 .. 7 : . 7 5
ð7Þ
oM m opn
As an example, we have applied the GLPU to the simple model z f(x,y) = 0. Following the formulation established above, in this case the vector of the input quantities is P = (x,y)T and the output quantity is Q = (z). Therefore n = 2 and m = 1. Additionally, the input uncertainty matrix, formed by the standards and the mutual uncertainties of x and y, is known: u2p
¼
u2 ðxÞ
uðx; yÞ
uðx; yÞ
u2 ðyÞ
:
ð8Þ
Applying Eqs. (6) and (7) we obtained that Sq = (1) and that Sp = (of/ox of/oy)T. Therefore by applying Eq. (4), we achieved the so-called law of propagation of uncertainties (LPU) for two input quantities:
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
u2 ðzÞ ¼
2 2 of of u2 ðxÞ þ u2 ðyÞ ox oy of of þ2 uðx; yÞ: ox oy
283
ommended using a uniform pdf [17,19,20]. Then, its standard uncertainty is taken as ð9Þ
The generalization of this law to more than two input quantities is straightforward. The terms (of/ ox)2u2(x) and (of/oy)2u2(y) are the contributions of the input quantities x and y, to the square of the uncertainty of the output quantity z. It should be observed that in the case of a single output quantity, the GLPU reduces to the LPU, and the matrix formulation becomes unnecessary. There are two approaches to evaluate the standard uncertainties of the input quantities (diagonal elements of the input uncertainty matrix). The type A evaluation [17] applies only to quantities which are measured directly several times under repeatability conditions. Uncertainty of input quantities that are measured only once, those are evaluated from models that involve further quantities, or that are imported from other sources, should be evaluated by the type B method of evaluation [17]. In many cases, this type involves obtaining an uncertainty as the standard deviation of the probability density function (pdf) that is assumed to apply. For example, if a quantity X is assumed to vary uniformly within a given range of width dX, is rec-
dX uðX Þ ¼ pffiffiffiffiffi : 12
ð10Þ
4. Sensitivity vector uncertainty 4.1. Sensitivity vector models According to [16], the sensitivity vector e = (ex, ey, ez)T of a dual-beam interferometer can be expressed as e¼
2p ½^n1 ^n2 ; k
ð11Þ
where the unitary vectors n^1 and ^n2 stand for the direction of the illuminating rays on each point of the observed area. The vectors ^n1 , ^n2 and e are shown in Fig. 3. The vector ^n1 can be expressed through a model that includes the angles a1, b1 and c1 of this unitary vector with respect to the corresponding coordinates x, y and z (see Fig. 3). ^ where ^n1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hence, p ¼ cos a1^i þ cos bffi 1^j cos c1 k, cos a1 ¼ 1 cos2 b1 cos2 c1 . If the angles of ^n2 with respect to the coordinates x, y and z are, respectively, a2, b2 andc2 according to Eq. (11),
Fig. 3. Sensitivity vector e of a dual-beam interferometer.
284
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
the components of the sensitivity vector e of Fig. 3 can be written as ex ¼
ffi 2p hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 cos2 ðc1 þ Dc1 Þ cos2 ðb1 þ Db1 Þ k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii þ 1 cos2 ðc2 þ Dc2 Þ cos2 ðb2 þ Db2 Þ ; ð12aÞ
ey ¼
2p ½cos ðb1 þ Db1 Þ cos ðb2 þ Db2 Þ; k
ð12bÞ
ez ¼
2p ½ cos ðc1 þ Dc1 Þ þ cos ðc2 þ Dc2 Þ; k
ð12cÞ
where we included the quantities Dc1, Dc2 and Db2 to account for eventual errors in the collimation of the incident beams. Although in our experiment the estimates of Dc1, Dc2, Db1 and b2 were zero, their corresponding uncertainties were not. Since we used the symmetrical interferometer of Fig. 1, with collimated beams traveling in the x–z plane, we considered that b1 = b2 = 90 and that c1 = c2. Therefore, by evaluating Eqs. (12a–c) we obtained that the estimates of the components of the sensitivity vector of this interferometer were ex = 4p sen c1/k, ey = 0 and ez = 0. Estimates of the sensitivity vector components can be affected by errors in the collimation and errors in the alignment of the wavefronts used to illuminate the sample. The latter errors can shift the values of b1, b2, c1 and c2; and the former errors can affect the values of Dc1, Dc2, Db1 and Db2. If Dc1, Dc2, Db1 and Db2 are not zero, ey and ez, evaluated by Eqs. (12b–c) are not zero. This means that the collimation errors produce slightly divergent beams changing in turn the sensitivity of the interferometer. 4.2. Uncertainty of the input quantities According to the formulation established in Section 3, with Eqs. (12a–c) we built up a set of three measurement models that we represented compactly as M(p, e) = 0, where the 8-dimensional vector of input quantities p = (b1, b2, c1, c2, Dc1, Dc2, Db1, Db2)T is related to the vector of output
quantities e = (ex, ey, ez)T. Since the maximum reasonable error associated to the estimated value of k is just about 0.1 nm, we decided to neglect the contribution of k to the uncertainty of the sensitivity vector components. Therefore, k was not considered as an element of the input vector p. Because, we assumed that the input quantities were uncorrelated, the 8 · 8 input uncertainty matrix u2p was diagonal. Its diagonal terms were the squares of the standard uncertainties of the input quantities b1, b2, c1, c2, Dc1, Dc2, Db1 and Db2. c1 is the angle with respect to z of the symmetry axis of the beam 1 of the interferometer of Fig. 3. Its standard uncertainty was taken as equal to the standard deviation of the uniform probability density function (pdf) that we assumed to apply. The width of the pdf was assigned considering the maximum reasonable deviation of c1 with respect to its estimated value. This deviation is the alignment error. Thus, according to Eq. (10) and considering that the maximum reasonable alignment error was 1, we used 2ð 1 Þ uðc1 Þ ¼ pffiffiffiffiffi : 12
ð13Þ
Analogous equations to Eq. (13) were utilized to evaluate the uncertainties of c2, b1 and b2, considering for these input quantities the same maximum reasonable alignment error. Fig. 4(a) shows the transversal section of the beam 1 of the interferometer of Fig. 3. This impinges on an area Lx · Ly and it has collimation errors. Since ^n1 stands for the direction of the illuminating ray on each point of the observed area, on the plane x–z, the collimation error Dc1 was defined as the angle between the symmetry axis of the beam and the local vector ^n1 . Similarly, on the plane y–z, Db1 was the angle between the symmetry axis of the beam and the local vector ^n1 . As it is shown in Fig. 4(b), the collimation errors depend on the position (x, y) of the illuminated point. At point (Lx/2, Ly/2), the corresponding illuminating ray has the largest collimation errors. For any other point (x, y) of the illuminated surface, we estimated these errors as x Dc1;Lx Dc1 ¼ ; ð14aÞ ðLx =2Þ
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
285
Fig. 4. (a) Transversal section of the beam 1 of Fig. 3, impinging on an area Lx · Ly. This beam has collimation errors and therefore, it appears slightly divergent. The collimation errors at the borders are denoted as Dc1,Lx and Db1,Ly. (b) Change in the collimation error with the position of the illuminated point. It should be observed that Dc1,Lx > Dc1.
y Db1;Ly ; Db1 ¼ Ly =2
ð14bÞ
where Dc1,Lx and Db1,Ly stand for the collimation errors of the rays that illuminate the borders of the observed surface. During our experiment, we presumed an adequate collimation and therefore the estimates of Dc1, Dc2, Db1 and Db2 were zero. However, their uncertainties were not. The standard uncertainties of Dc1 and Db1 were obtained by Eq. (10) assuming uniform pdfs. The width of the pdf was determined considering the maximum reasonable deviations of Dc1 and of Db1 with respect to their corresponding estimates. These deviations were evaluated considering for the angles Dc1,Lx and Db1,Ly in Eqs. (14a–b), values of 0.1. Then, since Lx and Ly were both equal to 30 mm, we obtained
uðDc1 Þ ¼
2xð0:1Þ pffiffiffiffiffi =mm; 15 12
uðDb1 Þ ¼
2y ð0:1Þ pffiffiffiffiffi =mm: 15 12
ð15aÞ
ð15bÞ
Analogous equations to Eqs. (14a–b) and to Eqs. (15a–b) were utilized to evaluate the uncertainties of Dc2 and for Db2, considering for Dc2,Lx and for Db2,Ly, angles of 0.1.
4.3. Uncertainty of the components of e According to the formulation established in Section 3, we built up a 8 · 8 diagonal input uncertainty matrix u2p with the standard uncertainties of the input quantities (elements of the vector p = (b1, b2, c1, c2, Dc1, Dc2, Db1, Db2)T). The output quantities (elements of the vector e = (ex, ey, ez)T) were related to the input quantities through Eqs. (12a–c) that formed a set of three measurement models written compactly as M(p, e) = 0. Then, by applying the GLPU (Eq. (4)), we obtained the 3 · 3 output uncertainty matrix 2 2 3 u ðex Þ uðex ; ey Þ uðex ; ez Þ 6 7 u2e ¼ 4 uðex ; ey Þ u2 ðey Þ uðez ; ey Þ 5: ð16Þ uðex ; ez Þ
uðez ; ey Þ
u2 ðez Þ
The diagonal terms of this matrix are the squares of the standard uncertainties of the components of the sensitivity vector e, and the off-diagonal terms are their mutual uncertainties. The application of the GLPU yielded zero for the terms u(ex, ey) and u(ez, ey) of the output uncertainty matrix u2e . Fig. 5 shows the obtained whole-field values of u(ex), u(ey), u(ez), and u(ex, ez). It can be observed that, although the uncertainties of ey and ez are not negligible, the mutual uncertainty
286
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
Fig. 5. Standard uncertainty of: (a) ex, (b) ey and (c) ez and (d) mutual uncertainty of ex and ez.
of ex and ez is very small and therefore, we considered that it can be ignored. The relative standard uncertainty of ex, calculated as, along line y = 1 mm is shown in Fig. 6. It can be observed that the relative standard uncertainty of ex calculated using the incidence angles (c1 and c2) both equal to 49.4 was about 0.6%. It is useful to compare this datum with that evaluated for a Moire´ interferometer. Incidence angles of 49.4 are used in Moire´ interferometry to illuminate a grating of 1200 lines/mm that can be glued onto the surface of a sample. According to the data reported by [7], the relative standard uncertainty of ex of a Moire´ interferometer that uses a commercial grating of 1200 lines/mm, is about 0.4%. In Fig. 6, the relative standard uncertainties of ex calculated using the same data and input uncertainties but changing the incidence angles of the beams (c1 and c2) measured with respect to the specimen normal are also shown. It can be observed that the standard uncertainty of ex de-
Fig. 6. Relative standard uncertainty of ex along y = mm, calculated using for both incidence angles (c1 and c2) the values indicated in the plot.
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
pended strongly on the angles of the incident wavefronts. Moreover, the spatial variation of this uncertainty along line y = 1 mm was small. 4.4. Nonlinearity problems Since Eq. (12a) and (12b) use some common quantities (b1, b2, Db1 and Db1), ex and ey are correlated. However, the mutual uncertainty u(ex, ey), obtained in Section 4.3 by applying the GLPU, was zero. This was not a reasonable result and it was due to the strong non-linearity of Eqs. (12a– c). Nevertheless, since the mutual uncertainty of ex and ez was small (see Fig. 5(d)), the mutual uncertainty of ex and ey was also presumed small and therefore we considered that the correlation of the components of the sensitivity vector can be ignored. Another nonlinearity problem was that linked with the terms cos2(b1 + Db1) and cos2(b2 + Db2) in Eq. (12a). Since the estimates of b1 andb2 were 90 and the estimates of Db1 and Db2 were zero, when the GLPU was applied, the contributions of b1, b2, Db1 and Db2 to the uncertainty of ex were zero irrespective of the uncertainties associated to these input quantities. This was not a reasonable result, and it was due to the strong nonlinearity of the square of the cosine function in the vicinity of 90. In order to evaluate the lost contributions of the quantities b1, b2, Db1 and Db2, to the standard uncertainty of ex, we decided to consider the term cos2(b1 + Db1) in Eq. (12a) as a single input quantity that we labeled as S1. Similarly, S2 ” cos2(b2 + Db2). In order to evaluate the standard uncertainty of the new input quantity S1, first, we reduced the term cos2(b1 + Db1): S 1 cos2 ðb1 þ Db1 Þ ¼ 12½1 þ R1 ;
287
tively, at b1 = 90 and at Db1 = 0. The widths of these pdfs were determined considering the maximum reasonable deviation angles with respect to their estimates. For Db1, this deviation was estimated using Eq. (14b); it gave 0.006.y. For b1, we used 1. Therefore 1 1 E½R21 ¼ 0:012 :y 2 Z 0006 :y Z 91 cos2 ð2b1 þ 2Db1 Þ db1 dDb1 ; 0006 :y
89
ð19aÞ 1 1 E½R1 ¼ 0:012 :y 2 Z 0006 :y Z 91 cos ð2b1 þ 2Db1 Þ db1 dDb1 :
0006 :y
89
ð19bÞ E½R21
and E[R1] evaluated by Eqs. (19a) and (19b) allowed us to estimate u2(R1) by Eq. (18). Since Eq. (17) is a single measurement model, the matrix formulation of Section 3 was unnecessary and the standard uncertainty of S1 was calculated by applying the LPU (Eq. (9–17)). The results are shown in Fig. 7 and they can be considered also valid for. Although the scale of the plot was selected in order to show the spatial variations of u(S1), these variations were relatively small. Computing the spatial average of the data plotted in Fig. 7, we concluded that the standard uncertainty
ð17Þ
where R1 = cos(2b1 + 2Db1). The application of the law of propagation of uncertainties to Eq. (17) allowed us to evaluate the uncertainty of S1 which in turn depended on the uncertainty of R1. According to the procedure recommended by [19], the latter was evaluated by 2
u2 ðR1 Þ ¼ E½R21 ½E½R1 ;
ð18Þ
where E is the expectation operator. We assumed for b1 and for Db1 uniform pdfs centered, respec-
Fig. 7. Standard uncertainty of S1 ” cos2(b1 + Db1).
288
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
of was just about 9.2 · 105. This means that the combined contribution of the input quantities b1 and Db1 to the standard uncertainty of ex should be very small. The same is valid for the contributions of b2 and Db2. Therefore, we concluded that the uncertainty evaluation performed in Section 4.3 was not affected significantly by the application of the GLPU to the non-linear equations (12a–c). Nevertheless, the values of u(ex) shown in Fig. 5(a) include the combined contribution of b1, Db1, b2 and Db2, calculated through the evaluation of u(S1) and u(S2) following the procedure described above. 4.5. Contributions to the ex uncertainty Fig. 8 depicts the different contributions to the square of u(ex) along line y = 1 mm, as a
function of the square of x. For clarity, these contributions have been labeled as C2(X) ” (oex/ oX)2u2(X) for generic input quantity X. The scale of the plot was selected in order to show the very small contributions of and, that in turn stand for the combined contributions of b1, Db1, b2 and Db2. It may be seen that the contributions of c1 and c2 are greater than those of Dc1 and Dc2. This means that eventual errors in the alignment have more influence on the uncertainty than faults in the collimation. Therefore, we conclude that, if a dual-beam interferometer is used, special attention must be paid to the accurate alignment of the incident beams. Moreover, since in Fig. 8 the contributions of Dc1 and Dc2 increase with the spatial coordinate x, we conclude that the effect of the collimation errors was greater on the illuminated area close to the borders. This means that the relative influence of the collimation errors on the uncertainty should increase with the size of the field.
5. Displacement uncertainty 5.1. Displacement model During a mechanical tensile test with load application along x, although the main relative displacement of the specimen should be observed along that direction, the sample dimension change along x induces also deformations along coordinates y and z. In this case, the three normal strains can be related by using the Poisson rate v [21]: ey ¼ ez ¼ vex ;
Fig. 8. Contributions to u2(ex) as a function of x2, along y = 1 mm. Line 1: C2(c1) + C2(c2); Line 2: C2(c1) + C2(c2) + C2[S1] + C2[S2]; Line 3: C2(c1) + C2(c2) + C2[S1] + C2[S2] + C2(Dc1) + C2(Dc2).
ð20Þ
where ex = oU/ox, ey = oV/oy and ez = oW/oz. As it can be observed in Fig. 2(a), we located both, the reference system utilized to determine the spatial coordinates of the field and the reference system used to evaluate the displacements, at the center of the illuminated area. Moreover, Fig. 2(b) has shown that the local displacement increased almost linearly with x. Therefore, for values of x and y different of zero, the strains associated to the local displacement components could be approximately evaluated by:
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
ex ¼
U ; x
ð21aÞ
ey ¼
V ; y
ð21bÞ
ez ¼
WT ; Lz
ð21cÞ
where, the total displacement induced along z(WT) and the sample thick (Lz) were used to evaluate ez. Combining Eqs. (1) and (20) and Eqs. (21a–c), and solving for U, we obtain U¼
D/x : ex x vey y vez Lz
ð22Þ
Since the components of the sensitivity vector of our ESPI interferometer were ex = 4p sen c1/k, ey = 0 and ez = 0, Eq. (22) reduces to Eq. (2). However, because the uncertainties of ey and ez were not negligible (see Fig. 5), we considered that Eq. (2) does not include all the uncertainty sources. Therefore, we took Eq. (22) as the measurement model to evaluate the displacement uncertainty by applying the law of propagation of uncertainties. 5.2. Uncertainty of the input quantities According to the formulation established in Section 3, Eq. (22) is a single measurement model where U is related to the input quantities D/, ex, ey and ez. Since the estimates of ey and ez were zero, when the LPU (Eq. (9)) was applied to Eq. (22), the contributions to the U uncertainty of the terms v and Lz were zero. Although it has been recommended several methods to include these lost contributions [19], independently of the selected technique, these contributions are very small compared to those of D/ and ex. Therefore, we decided ignore the standard uncertainties of v and Lz ; they were not considered as input quantities. As pointed out in Section 4.4, although the components of the sensitivity vector are correlated, in our case, their mutual uncertainties can be ignored. Therefore, we applied the law of propagation of uncertainties to the model (22) assuming that the input quantities were uncorrelated.
289
The standard uncertainty of D/ depends on systematic effects linked with the used phase-shifting procedure and on the influence of noise and the perturbing environment. Assuming that the first one was compensated adequately, we have considered just the eventual variations in the phase-difference caused by the optical noise and the external influence. We estimated a maximum eventual error of about 0.5 rad in the measured D/ values. Therefore, assuming a uniform pdf, we took 2ð0:5Þ uðD/Þ ¼ pffiffiffiffiffi rad: 12
ð23Þ
The standard uncertainty of D/, estimated by Eq. (23), was 0.28 rad. This is a standard uncertainty considerably greater than that reported for the D/ values measured under similar experimental conditions with a Moire´ interferometer of the same sensitivity of our ESPI interferometer. According to [7], for the D/ values measured by phase-shifting Moire´ interferometry, u( D/) is just about 0.13 rad. The difference can be explained by considering the different signal-to-noise ratios of ESPI and of Moire´ interferometry. Since ESPI yields patterns noisier than those obtained by Moire´, we speculate that in the former method, the influence of optical noise should be greater than in the case of Moire´. The standard uncertainties of ex,ey and ez were evaluated in Section 4.3. Fig. 5 shows these standard uncertainties. 5.3. Standard uncertainty of the displacement The standard uncertainty of U, at each point of the illuminated field, was calculated by applying the LPU (Eq. (9–22)). We used as input uncertainties the data plotted in Fig. 5, and the outcome of Eq. (23). Fig. 9 shows the whole-field results. Although the scale of the plot was selected in order to shown the spatial variations of the local displacement uncertainties, these variations were relatively small. Computing the spatial average of the data plotted in Fig. 9, we concluded that the displacement uncertainty associated to our measurements was about 0.019 lm. The uncertainties associated to the displacement measurements performed by an ESPI
290
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
Fig. 9. (a) Standard uncertainty map of U. (b) Contour diagram of (a).
interferometer are considerably greater than those reported for similar displacements when a Moire´ interferometer of the same sensitivity was used. Based on the data reported by [7], the uncertainty associated to a relative displacement of 1 lm, measured by using a Moire´ interferometer, is about 0.009 lm. The difference observed when it is compared this datum with the uncertainties plotted in Fig. 9, can be explained considering that the influence of the optical noise on speckle-based methods is greater than on Moire´ techniques. 5.4. Contributions to the displacement uncertainty Fig. 10 depicts the different contributions to the square of u(U) along line x = 1 mm, as a function of the square of U, where again C2(X) denotes the contribution of generic quantity X. It may be seen that the most important contributor to the displacement standard uncertainty was the corresponding phase-difference. Since the closely linear relationship between the displacement and the spatial coordinate x (see Fig. 2(b)), the contribution of ex increased almost linearly with the squares of the corresponding displacements. Although, as shown above, the uncertainties of ey and ez were not negligible, their contributions to the displacement uncertainty were imperceptible. This means that the changes of the nominally null sensitivity vector
Fig. 10. Contributions to u2(U) along y = 1 mm, as a function of U2 Line 1: C2(D/) Line 2: C2(D/) + C2(ex).
components, caused by faults in the collimation, can be ignored. Hence, since the expected errors in metrological applications of the speckle-based interferometric
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
techniques are generally smaller than the assumed maximum reasonable errors in the alignment and in the collimation, we concluded that reliable uncertainty evaluations of measurements obtained by an ESPI interferometer with collimated illumination, can be performed by applying the LPU to the simple model: U¼
kD/ : 2p½senðc1 þ Dc1 Þ þ senðc2 þ Dc2 Þ
ð24Þ
291
displacement uncertainty can be neglected if the maximum reasonable errors in alignment and in collimation are in the order of those assumed in this work. Since the expected errors in metrological applications of the speckle-based interferometric techniques are generally smaller than the assumed maximum reasonable errors, we conclude that the changes of the ESPI interferometer sensitivity, caused by faults in the collimation, can be generally ignored.
6. Summary and conclusions
Acknowledgements
An investigation was carried out to evaluate the uncertainty of displacements measured by using a dual-beam ESPI interferometer that used collimated illuminating beams. Displacements were induced by applying tensile load to a metallic sheet sample. The interferometer was sensible just along the pulling direction. The other two sensitivity vector components were nominally zero. Displacement measurements depend on the interferometer sensitivity that is affected by errors in the collimation and in the alignment of the illuminating beams. Collimation errors produce slightly divergent beams changing in turn the interferometer sensitivity. In this work, special attention was paid to evaluate the contributions to the displacement uncertainty of eventual changes of the interferometer sensitivity produced by faults in the collimation of the beams. We found that the uncertainty of the sensitivity vector components depended strongly on the angle of the incident wavefronts, measured with respect to the specimen normal. We established that the errors in the alignment had more influence on the uncertainty evaluation than the faults in the collimation. Moreover, we found that the effect of the collimation errors was greater on the illuminated area close to the borders. This means that the relative influence of the collimation errors on the uncertainty should increase with the size of the field. Although the uncertainties of the nominally null sensitivity vector components were not insignificant, we found that their contributions to the
R.R. Cordero thanks support of MECESUP PUC/19903 Project and VLaamse Interuniversitaire Road (VLIR IUC, Componente 6). A. Martı´nez and R. Rodrı´guez-Vera thank Consejo Nacional de Ciencia y Tecnologı´a (CONACYT) and Consejo de Ciencia y Tecnologı´a del Estado de Guanajuato (CONCYTEG) for their suport to carry out this research. We thank Dr. Gonzalo Fuster, Dr. Luciano Laroze and the referee, for helpful suggestions to the final draft of this paper.
References [1] R. Jones, C. Wykes, Holographic and Speckle Interferometry, first ed., Cambridge University Press, Cambridge, 1983. [2] D. Post, B. Han, P. Ifju, High Sensitivity Moire´: Experimental Analysis for Mechanics and Materials, Springer, New York, 1994. [3] J.M. Huntley, in: P.K. Rastogi (Ed.), Digital Speckle Pattern Interferometry and Related Techniques, Wiley, Chichester, 2001. [4] M. Takeda, H. Ina, S. Kobayashi, J. Opt. Soc. Am. 72 (1) (1982) 156160. [5] D.W. Robinson, G.T. Reid, Phase Unwrapping Methods. Interferogram Analysis, Institute of Physics, Bristol, UK, 1993. [6] M.R. Miller, I. Mohammed, P.S. Ho, Opt. Lasers Eng. 36 (2) (2001) 127. [7] R.R. Cordero, I. Lira, Opt. Commun. 237 (2004) 25. [8] J. Schwider, R. Burow, K.E. Elner, J. Grzanna, R. Spolaczyk, K. Merkel, Appl. Opt. 22 (1983) 3421. [9] J. Schmit, K. Creath, Appl. Opt. 34 (1995) 3610. [10] H. Zhang, M. Lalor, D. Burton, Opt. Lasers Eng. 31 (1999) 381. [11] K. Creath, J. Schmit, Opt. Laser Eng. 24 (1996) 365. [12] P.J. de Groot, L.L. Deck, Appl. Opt. 35 (1996) 2172.
292
R.R. Cordero et al. / Optics Communications 241 (2004) 279–292
[13] D. Albrecht, Opt. Lasers Eng. 31 (1) (1999) 6381. [14] H.J. Puga, R. Rodriguez-Vera, A. Martinez, Opt. Lasers Technol. 34 (1) (2002) 8192. [15] A. Martinez, R. Rodriguez-Vera, J.A. Rayas, H.J. Puga, Opt. Lasers Eng. 39 (2003) 525. [16] A. Martinez, R. Rodriguez-Vera, J.A. Rayas, H.J. Puga, Opt. Commun. 223 (2003) 239. [17] ISO Guide to the Expression of Uncertainty in Measurement, Geneva, 1993.
[18] T. Kreis, Holographic Interferometry Principles and Methods, first ed., Akademic, Berlin, 1996. [19] I. Lira, Evaluating the Uncertainty of Measurement: Fundamentals and Practical Guidance, Institute of Physics Publishing, Bristol, 2002. [20] R.R. Cordero, P. Roth, Metrologia 41 (4) (2004) L22. [21] S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, third ed., McGraw-Hill International Editions, Singapore, 1970.